Wings Adam Ashley-Cooper and Joe Tomane, replacement forward Benn Robinson and fly-half Ben Foley also touched down for the Wallabies, who played to their traditional running strengths in the backs with Genia calling the shots.
Argentina scored tries through centre Marcelo Bosch and scrumhalf Martin Landajo with fly-half Nicolas Sanchez kicking seven points.
New Zealand retained the title by beating South Africa 38-27 earlier on Saturday.
Australia centre Christian Leali'ifano kicked 10 points and replacement fly-half Foley scored nine with a try and two conversions. Genia, who started the last two Tests on the bench, was in imperious form on his return to the starting line-up.
"It's a real confidence booster for us ... We're very, very happy with the result," Australia's captain James Horwill said.
"Sometimes you get the bounce of the ball and things come off ... 1-15 really put in the effort, we're really pleased to get the result, especially over here in Rosario."
Australia, who had beaten Argentina 14-13 in Perth last month, had forwards James Slipper and Rob Simmons sin-binned, one in each half, while Argentina flanker Pablo Matera was also yellow carded.
Pumas captain Juna Fernandez Lobbe gave credit to the Wallabies for their performance.
"We are disappointed, also we have to congratulate Australia the way they played," he said.
"They came here (backs) against the wall and played great rugby ... we missed too many tackles."
The Wallabies came hurtling out of the blocks and within three minutes they had crossed the Pumas line, Folau scoring his first try from Tomane's layoff in the tackle, with Leal'ifano converting.
Argentina pinned down the Wallabies pack five metres from their line through repeated scrums, with prop Slipper earning a yellow card, but the Pumas failed to take advantage
A Leali'ifano penalty put Australia 10-0 up before Sanchez hit back for the Pumas with a penalty under the posts after an infringement by Horwill.
Argentina flanker Matera went to the sin-bin for the last 10 minutes of the half for an infringement preventing Horwill from breaking through for a try with Australia getting another penalty instead, put over by Leali'ifano.
Australia then hit Argentina with two tries in two minutes past the half-hour mark, wing Ashley-Cooper touching down in the corner after a counter-attack from a lineout.
Folau then scored his second try immediately from the restart, breaking through a gap in the middle of the Argentine defence to pad the Wallabies lead to 25-3.
The Pumas then put together an attack across the width of the pitch to stretch the Wallabies' defence and centre Bosch broke through the middle to touch down under the posts.
Australia stayed on the pitch during halftime and began the second half as they had started the first, attacking through several phases before Folau collected his third try.
Argentina scored their second try after a break by left wing Juan Imhoff, who was brought down just before the line and Landajo squeezed over from the resulting ruck.
Wallabies lock Simmons was sin-binned in the 50th minute for an off-the-ball body check on centre Felipe Contepomi, who was likely making his final international appearance in a Pumas record 87th Test.
Tomane added a late breakaway try for the Wallabies after a passing error in the Pumas backs before Foley converted and then scored another breakaway try at the death, which he also converted to complete the scoring.